640 
R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-10: Part 1 
Ocean Sci., 6, 633-677, 2010 
www.ocean-sci.net/6/633/2010/ 
dry_f_si. The Helmholtz function for air-vapour interac- where v=Hp is the specific volume. Therefore, the first 
tion, derivatives of / F give the specific entropy, rj, 
f ma (A,T,p)=2A(l-A)p 
RT 
MaMw 
(2.13) 
Baw(T) + -p 
'A (1 — A) 
-r- Caaw(T) H — Caww(T) 
Ma Mw 
/t 
and the absolute pressure, P, 
(3.2) 
together with its partial derivatives is implemented as the 
library function air_f_mix_si. The cross-virial coef 
ficients Baw, Caaw and Caww are implemented as the 
library functions air_baw_m3mol, air_caaw_m6mol2 
and air_caww_m6mol2. The Helmholtz function of hu 
mid air, / AV (A,T, p), Eq. (2.7), together with its partial 
derivatives is implemented as the library function air_f_si. 
For convenience of use, some auxiliary conversion functions, 
Eqs. (2.9-2.12), are also implemented at level 0, Table SI. 
Deviating from the original formulation given by Lemmon 
et al. (2000), in the library the adjustable constants of dry 
air are specified such that the entropy and the enthalpy of 
dry air are zero at the standard ocean state, T=273.15 K and 
P=101325Pa (Feistel et al., 2010a). This choice does not 
affect any measurable thermodynamic properties. 
P 
df_ 
dp 
— „2 
= P fp- 
(3.3) 
A list of properties derived from f F (T,p) by means of 
Eqs. (3.2) and (3.3) is given in Table S2. Partial derivatives 
with respect to these two independent variables are written 
as subscripts. Whether the property belongs to liquid water 
or vapour depends on the density used, i.e. on the location in 
the diagram in Fig. 1. 
3.2 Ice 
The total differential of the Gibbs function g ,h (T. P) of ice 
Ih has the form 
dg Ih = — rjdT + vdP. (3.4) 
3 Level 2: Directly derived properties 
Its first derivatives give the specific entropy, rj, 
From the level-one functions described in Sect. 2, various 
thermodynamic properties can be computed directly if the 
corresponding independent variables are known. If some of 
the input variables need to be derived first from other known 
ones, based on thermodynamic relations, then the function 
will be found at level 3 (Sect. 4) or higher. 
The required input variables for level 2 functions are 
temperature and density of fluid pure water, either liq 
uid or vapour (Sect. 3.1), temperature and pressure for ice 
(Sect. 3.2), and Absolute Salinity, temperature and pressure 
for dissolved sea salt (Sect. 3.3). For moist air, level 2 rou 
tines require inputs of temperature, density and the mass 
fraction of (dry) air in the mixture. Specifying the air mass 
fraction as 1 gives the dry air limit. 
The Jacobi method developed by Shaw (1935) is the math 
ematically most elegant way of transforming the various 
partial derivatives of different potential functions into each 
other, exploiting the convenient formal calculus of functional 
determinants (Margenau and Murphy, 1943; Landau and Lif- 
schitz, 1964). Conversion tables (Feistel, 2008) between the 
potentials f(T, p), g (Sa, T, P) and h (Sa, P, P) are given in 
Sect. 5. 
3.1 Fluid water 
The total differential of the Helmholtz function / F (T,p) of 
fluid water has the form 
d/ F = — r) dT — Pdv — — rj dT H -dp, (3.1) 
P 2 
• I 
= ~8t 
(3.5) 
and the specific volume, v, 
Vf 
dP 
= 8p- 
(3.6) 
A list of properties derived from g lh (T, P) is given in Ta 
ble S3. Partial derivatives with respect to the two indepen 
dent variables are written as subscripts. 
3.3 Dissolved sea salt 
The total differential of the saline part g s (Sa, T, P) of the 
Gibbs function of seawater has the form 
dg s = — p s dT + v s dP + f.id^A■ 
(3.7) 
Its first derivatives give the saline part of the specific entropy, 
rj S , 
r> S = 
d Jl 
dT 
= ~8t> 
(3.8) 
S,P 
the saline part of the specific volume, n s , 
u 5 = 
dP 
S,T 
= 8p, 
(3.9)